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Intransitive dice
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<h2 id="example">Example</h2> <p>Consider the following set of dice. </p> <ul><li>Die <i>A</i> has sides 2, 2, 4, 4, 9, 9.</li> <li>Die <i>B</i> has sides 1, 1, 6, 6, 8, 8.</li> <li>Die <i>C</i> has sides 3, 3, 5, 5, 7, 7.</li></ul> <p>The <a href="/facts/Probability/fgYvMhwb">probability</a> that <i>A</i> rolls a higher number than <i>B</i>, the probability that <i>B</i> rolls higher than <i>C</i>, and the probability that <i>C</i> rolls higher than <i>A</i> are all 5/9, so this set of dice is intransitive. In fact, it has the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time. </p><p>Now, consider the following game, which is played with a set of dice. </p> <ol><li>The first player chooses a die from the set.</li> <li>The second player chooses one die from the remaining dice.</li> <li>Both players roll their die; the player who rolls the higher number wins.</li></ol> <p>If this game is played with a transitive set of dice, it is either fair or biased in favor of the first player, because the first player can always find a die that will not be beaten by any other dice more than half the time. If it is played with the set of dice described above, however, the game is biased in favor of the second player, because the second player can always find a die that will beat the first player's die with probability 5/9. The following tables show all possible outcomes for all three pairs of dice. </p> <table><tbody><tr><td colspan="4">Player 1 chooses die APlayer 2 chooses die C</td><td rowspan="5"></td><td colspan="4">Player 1 chooses die BPlayer 2 chooses die A</td><td rowspan="5"></td><td colspan="4">Player 1 chooses die CPlayer 2 chooses die B</td></tr><tr><th>AC</th><th>2</th><th>4</th><th>9</th><th>BA</th><th>1</th><th>6</th><th>8</th><th>CB</th><th>3</th><th>5</th><th>7</th></tr><tr><th>3</th><td>C</td><td>A</td><td>A</td><th>2</th><td>A</td><td>B</td><td>B</td><th>1</th><td>C</td><td>C</td><td>C</td></tr><tr><th>5</th><td>C</td><td>C</td><td>A</td><th>4</th><td>A</td><td>B</td><td>B</td><th>6</th><td>B</td><td>B</td><td>C</td></tr><tr><th>7</th><td>C</td><td>C</td><td>A</td><th>9</th><td>A</td><td>A</td><td>A</td><th>8</th><td>B</td><td>B</td><td>B</td></tr></tbody></table> <p>If one allows weighted dice, i.e., with unequal probability weights for each side, then alternative sets of three dice can achieve even larger probabilities than 5 9 ≈ 0.56 {\displaystyle {\frac {5}{9}}\approx 0.56} that each die beats the next one in the cycle. The largest possible probability is one over the <a href="/facts/Golden_ratio/anloSRmD">golden ratio</a>, 1 φ ≈ 0.62 {\displaystyle {\frac {1}{\varphi }}\approx 0.62} .<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> <h2 id="variations">Variations</h2> <h3>Efron's dice</h3> <p>Efron's dice are a set of four intransitive dice invented by <a href="/facts/Bradley_Efron/D0f95TyH">Bradley Efron</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <p>The four dice A, B, C, D have the following numbers on their six faces: </p> <ul><li>A: 4, 4, 4, 4, 0, 0</li> <li>B: 3, 3, 3, 3, 3, 3</li> <li>C: 6, 6, 2, 2, 2, 2</li> <li>D: 5, 5, 5, 1, 1, 1</li></ul> <p>Each die is beaten by the previous die in the list with wraparound, with probability 2/3. C beats A with probability 5/9, and B and D have equal chances of beating the other.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> If each player has one set of Efron's dice, there is a continuum of optimal strategies for one player, in which they choose their die with the following probabilities, where 0 ≤ <i>x</i> ≤ 3/7:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> P(choose A) = <i>x</i> P(choose B) = 1/2 - 5/6<i>x</i> P(choose C) = <i>x</i> P(choose D) = 1/2 - 7/6<i>x</i> <h3>Miwin's dice</h3> <p class="note">Main article: <a href="/facts/Miwin%2527s_dice/nYSeSFH7">Miwin's dice</a></p> <p>Miwin's Dice were invented in 1975 by the physicist Michael Winkelmann. </p><p>Consider a set of three dice, III, IV and V such that </p> <ul><li>die III has sides 1, 2, 5, 6, 7, 9</li> <li>die IV has sides 1, 3, 4, 5, 8, 9</li> <li>die V has sides 2, 3, 4, 6, 7, 8</li></ul> <p>Then: </p> <ul><li>the <a href="/facts/Probability/fgYvMhwb">probability</a> that III rolls a higher number than IV is 17/36</li> <li>the probability that IV rolls a higher number than V is 17/36</li> <li>the probability that V rolls a higher number than III is 17/36</li></ul> <h2 id="intransitive-dice-set-for-more-than-two-players">Intransitive dice set for more than two players</h2> <p>A number of people have introduced variations of intransitive dice where one can compete against more than one opponent. </p> <h3>Three players</h3> <h4>Oskar dice</h4> <p><a href="/facts/Oskar_van_Deventer/pMtDlKxI">Oskar van Deventer</a> introduced a set of seven dice (all faces with probability 1/6) as follows:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p> <ul><li>A: 2, 02, 14, 14, 17, 17</li> <li>B: 7, 07, 10, 10, 16, 16</li> <li>C: 5, 05, 13, 13, 15, 15</li> <li>D: 3, 03, 09, 09, 21, 21</li> <li>E: 1, 01, 12, 12, 20, 20</li> <li>F: 6, 06, 08, 08, 19, 19</li> <li>G: 4, 04, 11, 11, 18, 18</li></ul> <p>One can verify that A beats {B,C,E}; B beats {C,D,F}; C beats {D,E,G}; D beats {A,E,F}; E beats {B,F,G}; F beats {A,C,G}; G beats {A,B,D}. Consequently, for arbitrarily chosen two dice there is a third one that beats both of them. Namely, </p> <ul><li>G beats {A,B}; F beats {A,C}; G beats {A,D}; D beats {A,E}; D beats {A,F}; F beats {A,G};</li> <li>A beats {B,C}; G beats {B,D}; A beats {B,E}; E beats {B,F}; E beats {B,G};</li> <li>B beats {C,D}; A beats {C,E}; B beats {C,F}; F beats {C,G};</li> <li>C beats {D,E}; B beats {D,F}; C beats {D,G};</li> <li>D beats {E,F}; C beats {E,G};</li> <li>E beats {F,G}.</li></ul> <p>Whatever the two opponents choose, the third player will find one of the remaining dice that beats both opponents' dice. </p> <h4>Grime dice</h4> <p>Dr. James Grime discovered a set of five dice as follows:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <ul><li>A: 2, 2, 2, 7, 7, 7</li> <li>B: 1, 1, 6, 6, 6, 6</li> <li>C: 0, 5, 5, 5, 5, 5</li> <li>D: 4, 4, 4, 4, 4, 9</li> <li>E: 3, 3, 3, 3, 8, 8</li></ul> <p>One can verify that, when the game is played with one set of Grime dice: </p> <ul><li>A beats B beats C beats D beats E beats A (first chain);</li> <li>A beats C beats E beats B beats D beats A (second chain).</li></ul> <p>However, when the game is played with two such sets, then the first chain remains the same, except that D beats C, but the second chain is reversed (i.e. A beats D beats B beats E beats C beats A). Consequently, whatever dice the two opponents choose, the third player can always find one of the remaining dice that beats them both (as long as the player is then allowed to choose between the one-die option and the two-die option): </p> <table><tbody><tr><th colspan="2" rowspan="2">Sets chosenby opponents</th><th colspan="2">Winning set of dice</th></tr><tr><th>Type</th><th>Number</th></tr><tr><td>A</td><td>B</td><td>E</td><td>1</td></tr><tr><td>A</td><td>C</td><td>E</td><td>2</td></tr><tr><td>A</td><td>D</td><td>C</td><td>2</td></tr><tr><td>A</td><td>E</td><td>D</td><td>1</td></tr><tr><td>B</td><td>C</td><td>A</td><td>1</td></tr><tr><td>B</td><td>D</td><td>A</td><td>2</td></tr><tr><td>B</td><td>E</td><td>D</td><td>2</td></tr><tr><td>C</td><td>D</td><td>B</td><td>1</td></tr><tr><td>C</td><td>E</td><td>B</td><td>2</td></tr><tr><td>D</td><td>E</td><td>C</td><td>1</td></tr></tbody></table> <h3>Four players</h3> <p>It has been proved that a four player set would require at least 19 dice.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> In July 2024 <a href="/facts/GitHub/n1hNCsc3">GitHub</a> user NGeorgescu published a set of 23 eleven sided dice which satisfy the constraints of the four player intransitive dice problem.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> The set has not been published in an academic journal or been peer-reviewed. </p> <h3>Four players</h3> <p>A four-player set is proved to require at least 19 dice.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h4>Georgescu dice</h4> <p>In 2024, American scientist Nicholas S. Georgescu discovered a set of 23 dice which solve the four-player intransitive dice problem.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> <table><tbody><tr><td>0</td><td>40</td><td>61</td><td>83</td><td>105</td><td>116</td><td>158</td><td>173</td><td>203</td><td>213</td><td>234</td></tr><tr><td>1</td><td>29</td><td>46</td><td>89</td><td>109</td><td>119</td><td>153</td><td>175</td><td>196</td><td>226</td><td>243</td></tr><tr><td>2</td><td>41</td><td>54</td><td>72</td><td>113</td><td>122</td><td>148</td><td>177</td><td>189</td><td>216</td><td>252</td></tr><tr><td>3</td><td>30</td><td>62</td><td>78</td><td>94</td><td>125</td><td>143</td><td>179</td><td>205</td><td>229</td><td>238</td></tr><tr><td>4</td><td>42</td><td>47</td><td>84</td><td>98</td><td>128</td><td>138</td><td>181</td><td>198</td><td>219</td><td>247</td></tr><tr><td>5</td><td>31</td><td>55</td><td>90</td><td>102</td><td>131</td><td>156</td><td>183</td><td>191</td><td>209</td><td>233</td></tr><tr><td>6</td><td>43</td><td>63</td><td>73</td><td>106</td><td>134</td><td>151</td><td>162</td><td>184</td><td>222</td><td>242</td></tr><tr><td>7</td><td>32</td><td>48</td><td>79</td><td>110</td><td>137</td><td>146</td><td>164</td><td>200</td><td>212</td><td>251</td></tr><tr><td>8</td><td>44</td><td>56</td><td>85</td><td>114</td><td>117</td><td>141</td><td>166</td><td>193</td><td>225</td><td>237</td></tr><tr><td>9</td><td>33</td><td>64</td><td>91</td><td>95</td><td>120</td><td>159</td><td>168</td><td>186</td><td>215</td><td>246</td></tr><tr><td>10</td><td>45</td><td>49</td><td>74</td><td>99</td><td>123</td><td>154</td><td>170</td><td>202</td><td>228</td><td>232</td></tr><tr><td>11</td><td>34</td><td>57</td><td>80</td><td>103</td><td>126</td><td>149</td><td>172</td><td>195</td><td>218</td><td>241</td></tr><tr><td>12</td><td>23</td><td>65</td><td>86</td><td>107</td><td>129</td><td>144</td><td>174</td><td>188</td><td>208</td><td>250</td></tr><tr><td>13</td><td>35</td><td>50</td><td>69</td><td>111</td><td>132</td><td>139</td><td>176</td><td>204</td><td>221</td><td>236</td></tr><tr><td>14</td><td>24</td><td>58</td><td>75</td><td>92</td><td>135</td><td>157</td><td>178</td><td>197</td><td>211</td><td>245</td></tr><tr><td>15</td><td>36</td><td>66</td><td>81</td><td>96</td><td>115</td><td>152</td><td>180</td><td>190</td><td>224</td><td>231</td></tr><tr><td>16</td><td>25</td><td>51</td><td>87</td><td>100</td><td>118</td><td>147</td><td>182</td><td>206</td><td>214</td><td>240</td></tr><tr><td>17</td><td>37</td><td>59</td><td>70</td><td>104</td><td>121</td><td>142</td><td>161</td><td>199</td><td>227</td><td>249</td></tr><tr><td>18</td><td>26</td><td>67</td><td>76</td><td>108</td><td>124</td><td>160</td><td>163</td><td>192</td><td>217</td><td>235</td></tr><tr><td>19</td><td>38</td><td>52</td><td>82</td><td>112</td><td>127</td><td>155</td><td>165</td><td>185</td><td>207</td><td>244</td></tr><tr><td>20</td><td>27</td><td>60</td><td>88</td><td>93</td><td>130</td><td>150</td><td>167</td><td>201</td><td>220</td><td>230</td></tr><tr><td>21</td><td>39</td><td>68</td><td>71</td><td>97</td><td>133</td><td>145</td><td>169</td><td>194</td><td>210</td><td>239</td></tr><tr><td>22</td><td>28</td><td>53</td><td>77</td><td>101</td><td>136</td><td>140</td><td>171</td><td>187</td><td>223</td><td>248</td></tr></tbody></table> <h4>Li dice</h4> <p>Youhua Li subsequently developed a set of 19 dice with 171 faces each that solves the four-player problem. This has been shown to be extensible for any number of dice given a domination graph with <i>n</i> nodes, producing dice with <i>n(n−1)/2</i> faces.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> </p> <h2 id="intransitive-12-sided-dice">Intransitive 12-sided dice</h2> <p>In analogy to the intransitive six-sided dice, there are also dodecahedra which serve as intransitive <a href="/facts/Dice/aURlPtbp">twelve-sided dice</a>. The points on each of the dice result in the sum of 114. There are no repetitive numbers on each of the dodecahedra. </p><p>Miwin's dodecahedra (set 1) win cyclically against each other in a ratio of 35:34. </p><p>The miwin's dodecahedra (set 2) win cyclically against each other in a ratio of 71:67. </p><p>Set 1: </p> <table><tbody><tr><td>D III</td><td>purple</td><td>1</td><td>2</td><td></td><td></td><td>5</td><td>6</td><td>7</td><td></td><td>9</td><td>10</td><td>11</td><td></td><td></td><td>14</td><td>15</td><td>16</td><td></td><td>18</td></tr><tr><td>D IV</td><td>red</td><td>1</td><td></td><td>3</td><td>4</td><td>5</td><td></td><td></td><td>8</td><td>9</td><td>10</td><td></td><td>12</td><td>13</td><td>14</td><td></td><td></td><td>17</td><td>18</td></tr><tr><td>D V</td><td>dark grey</td><td></td><td>2</td><td>3</td><td>4</td><td></td><td>6</td><td>7</td><td>8</td><td></td><td></td><td>11</td><td>12</td><td>13</td><td></td><td>15</td><td>16</td><td>17</td><td></td></tr></tbody></table> <p>Set 2: </p> <table><tbody><tr><td>D VI</td><td>cyan</td><td>1</td><td>2</td><td>3</td><td>4</td><td></td><td></td><td></td><td></td><td>9</td><td>10</td><td>11</td><td>12</td><td>13</td><td>14</td><td></td><td></td><td>17</td><td>18</td></tr><tr><td>D VII</td><td>pear green</td><td>1</td><td>2</td><td></td><td></td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td></td><td></td><td></td><td></td><td>15</td><td>16</td><td>17</td><td>18</td></tr><tr><td>D VIII</td><td>light grey</td><td></td><td></td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td></td><td></td><td>11</td><td>12</td><td>13</td><td>14</td><td>15</td><td>16</td><td></td><td></td></tr></tbody></table> <h3>Intransitive prime-numbered 12-sided dice</h3> <p>It is also possible to construct sets of intransitive dodecahedra such that there are no repeated numbers and all numbers are primes. Miwin's intransitive prime-numbered dodecahedra win cyclically against each other in a ratio of 35:34. </p><p>Set 1: The numbers add up to 564. </p> <table><tbody><tr><td>PD 11</td><td>grey to blue</td><td>13</td><td>17</td><td>29</td><td>31</td><td>37</td><td>43</td><td>47</td><td>53</td><td>67</td><td>71</td><td>73</td><td>83</td></tr><tr><td>PD 12</td><td>grey to red</td><td>13</td><td>19</td><td>23</td><td>29</td><td>41</td><td>43</td><td>47</td><td>59</td><td>61</td><td>67</td><td>79</td><td>83</td></tr><tr><td>PD 13</td><td>grey to green</td><td>17</td><td>19</td><td>23</td><td>31</td><td>37</td><td>41</td><td>53</td><td>59</td><td>61</td><td>71</td><td>73</td><td>79</td></tr></tbody></table> <p>Set 2: The numbers add up to 468. </p> <table><tbody><tr><td>PD 1</td><td>olive to blue</td><td>7</td><td>11</td><td>19</td><td>23</td><td>29</td><td>37</td><td>43</td><td>47</td><td>53</td><td>61</td><td>67</td><td>71</td></tr><tr><td>PD 2</td><td>teal to red</td><td>7</td><td>13</td><td>17</td><td>19</td><td>31</td><td>37</td><td>41</td><td>43</td><td>59</td><td>61</td><td>67</td><td>73</td></tr><tr><td>PD 3</td><td>purple to green</td><td>11</td><td>13</td><td>17</td><td>23</td><td>29</td><td>31</td><td>41</td><td>47</td><td>53</td><td>59</td><td>71</td><td>73</td></tr></tbody></table> <h2 id="generalized-muoz-pereras-intransitive-dice">Generalized Muñoz-Perera's intransitive dice</h2> <p>A generalization of sets of intransitive dice with N {\displaystyle N} faces is possible.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Given N ≥ 3 {\displaystyle N\geq 3} , we define the set of dice { D n } n = 1 N {\displaystyle \{D_{n}\}_{n=1}^{N}} as the random variables taking values each in the set { v n , j } j = 1 J {\displaystyle \{v_{n,j}\}_{j=1}^{J}} with </p><p> P [ D n = v n , j ] = 1 J {\displaystyle \mathbb {P} \left[D_{n}=v_{n,j}\right]={\frac {1}{J}}} , </p><p>so we have N {\displaystyle N} fair dice of J {\displaystyle J} faces. </p><p>To obtain a set of intransitive dice is enough to set the values v n , j {\displaystyle v_{n,j}} for n , j = 1 , 2 , … , N {\displaystyle n,j=1,2,\ldots ,N} with the expression </p><p> v n , j = ( j − 1 ) N + ( n − j ) mod ( N ) + 1 {\displaystyle v_{n,j}=(j-1)N+(n-j){\text{mod}}(N)+1} , </p><p>obtaining a set of N {\displaystyle N} fair dice of N {\displaystyle N} faces </p><p>Using this expression, it can be verified that </p><p> P [ D m < D n ] = 1 2 + 1 2 N − ( n − m ) mod ( N ) N 2 {\displaystyle \mathbb {P} \left[D_{m}<D_{n}\right]={\frac {1}{2}}+{\frac {1}{2N}}-{\frac {(n-m){\text{mod}}(N)}{N^{2}}}} , </p><p>So each die beats ⌊ N / 2 − 1 ⌋ {\displaystyle \lfloor N/2-1\rfloor } dice in the set. </p> <h3>Examples</h3> <h4>3 faces</h4> <table><tbody><tr><td> D 1 {\displaystyle D_{1}} </td><td>1</td><td>6</td><td>8</td></tr><tr><td> D 2 {\displaystyle D_{2}} </td><td>2</td><td>4</td><td>9</td></tr><tr><td> D 3 {\displaystyle D_{3}} </td><td>3</td><td>5</td><td>7</td></tr></tbody></table> <p>The set of dice obtained in this case is equivalent to the first example on this page, but removing repeated faces. It can be verified that D 3 > D 2 , D 2 > D 1 and D 1 > D 3 {\displaystyle D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{3}} . </p> <h4>4 faces</h4> <table><tbody><tr><td> D 1 {\displaystyle D_{1}} </td><td>1</td><td>8</td><td>11</td><td>14</td></tr><tr><td> D 2 {\displaystyle D_{2}} </td><td>2</td><td>5</td><td>12</td><td>15</td></tr><tr><td> D 3 {\displaystyle D_{3}} </td><td>3</td><td>6</td><td>9</td><td>16</td></tr><tr><td> D 4 {\displaystyle D_{4}} </td><td>4</td><td>7</td><td>10</td><td>13</td></tr></tbody></table> <p>Again it can be verified that D 4 > D 3 , D 3 > D 2 , D 2 > D 1 and D 1 > D 4 {\displaystyle D_{4}>D_{3},D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{4}} . </p> <h4>6 faces</h4> <table><tbody><tr><td> D 1 {\displaystyle D_{1}} </td><td>1</td><td>12</td><td>17</td><td>22</td><td>27</td><td>32</td></tr><tr><td> D 2 {\displaystyle D_{2}} </td><td>2</td><td>7</td><td>18</td><td>23</td><td>28</td><td>33</td></tr><tr><td> D 3 {\displaystyle D_{3}} </td><td>3</td><td>8</td><td>13</td><td>24</td><td>29</td><td>34</td></tr><tr><td> D 4 {\displaystyle D_{4}} </td><td>4</td><td>9</td><td>14</td><td>19</td><td>30</td><td>35</td></tr><tr><td> D 5 {\displaystyle D_{5}} </td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td><td>36</td></tr><tr><td> D 6 {\displaystyle D_{6}} </td><td>6</td><td>11</td><td>16</td><td>21</td><td>26</td><td>31</td></tr></tbody></table> <p>Again D 6 > D 5 , D 5 > D 4 , D 4 > D 3 , D 3 > D 2 , D 2 > D 1 and D 1 > D 6 {\displaystyle D_{6}>D_{5},D_{5}>D_{4},D_{4}>D_{3},D_{3}>D_{2},D_{2}>D_{1}\ {\text{and}}\ D_{1}>D_{6}} . Moreover D 6 > { D 5 , D 4 } , D 5 > { D 4 , D 3 } , D 4 > { D 3 , D 2 } , D 3 > { D 2 , D 1 } , D 2 > { D 1 , D 6 } and D 1 > { D 6 , D 5 } {\displaystyle D_{6}>\{D_{5},D_{4}\},D_{5}>\{D_{4},D_{3}\},D_{4}>\{D_{3},D_{2}\},D_{3}>\{D_{2},D_{1}\},D_{2}>\{D_{1},D_{6}\}\ {\text{and}}\ D_{1}>\{D_{6},D_{5}\}} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Blotto_games/74BL0611">Blotto games</a></li> <li><a href="/facts/Freivalds%2527_algorithm/wClqXGFR">Freivalds' algorithm</a></li> <li><a href="/facts/Go_First_Dice/v3xEH3mV">Go First Dice</a></li> <li><a href="/facts/Nontransitive_game/uTlbBG7D">Nontransitive game</a></li> <li><a href="/facts/Rock_paper_scissors/mkUcl9Nv">Rock paper scissors</a></li> <li>Condorcet's <a href="/facts/Voting_paradox/PXuWT0jl">voting paradox</a></li></ul> <h2 id="sources">Sources</h2> <ul><li><a href="/facts/Martin_Gardner/HGfEnPDH">Gardner, Martin</a> (2001). <a href="https://archive.org/details/colossalbookmath00gard"><i>The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics</i></a> (1st ed.). New York: W. W. Norton & Company. p. <a href="https://archive.org/details/colossalbookmath00gard/page/n302">286</a>–311.[<i>ISBN missing</i>]</li> <li><i>Spielerische Mathematik mit Miwin'schen Würfeln</i> (in German). Bildungsverlag Lemberger. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-85221-531-0.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://mathworld.wolfram.com/EfronsDice.html">MathWorld page</a></li> <li><a href="https://www.sciencenews.org/article/tricky-dice-revisited">Ivars Peterson's MathTrek - Tricky Dice Revisited (April 15, 2002)</a></li> <li><a href="https://web.archive.org/web/20061016165934/http://www.jimloy.com/puzz/nontran.htm">Jim Loy's Puzzle Page</a></li> <li><a href="http://www.miwin.com">Miwin official site</a> (in German)</li> <li><a href="https://web.archive.org/web/20120723135456/http://www.devrand.org/view/diceFinder">Open Source nontransitive dice finder</a></li> <li><a href="http://singingbanana.com/dice/article.htm">Non-transitive Dice by James Grime</a></li> <li><a href="http://www.mathsgear.co.uk/non-transitive-dice/">Maths Gear</a></li> <li><a href="https://www.maa.org/programs/maa-awards/writing-awards/intransitive-dice">Conrey, B., Gabbard, J., Grant, K., Liu, A., & Morrison, K. (2016). Intransitive dice. <i>Mathematics Magazine,</i> 89(2), 133-143. Awarded by Mathematical Association of America</a>[<i>permanent dead link</i>]</li> <li><a href="/facts/Timothy_Gowers/El3xoEmE">Timothy Gowers</a>' <a href="http://michaelnielsen.org/polymath1/index.php?title=Intransitive_dice">project on intransitive dice</a></li> <li><a href="/facts/Erica_Klarreich/arTCNGU4">Klarreich, Erica</a> (2023-01-19). <a href="https://www.quantamagazine.org/mathematicians-roll-dice-and-get-rock-paper-scissors-20230119/">"Mathematicians Roll Dice and Get Rock-Paper-Scissors"</a>. <i><a href="/facts/Quanta_Magazine/FkbYnGtU">Quanta Magazine</a></i>.</li> <li><a href="https://pereradrian.github.io">Adrián Muñoz Perera's site</a>'</li> <li>Introduction to Non-Transitive Gambling Bets for Magicians by Bruce Carlley. This is the ONLY book on this topic.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weisstein, Eric W. "Efron's Dice". Wolfram MathWorld. Retrieved 12 January 2021. <a href="/wiki/Eric_W._Weisstein" target="_blank">/wiki/Eric_W._Weisstein</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Bogomolny, Alexander. "Non-transitive Dice". Cut the Knot. Archived from the original on 2016-01-12. <a href="/wiki/Alexander_Bogomolny" target="_blank">/wiki/Alexander_Bogomolny</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Savage, Richard P. (May 1994). "The Paradox of Nontransitive Dice". The American Mathematical Monthly. 101 (5): 429–436. doi:10.2307/2974903. JSTOR 2974903. <a href="https://www.jstor.org/stable/2974903" target="_blank">https://www.jstor.org/stable/2974903</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021. <a href="https://www.jstor.org/stable/2690722" target="_blank">https://www.jstor.org/stable/2690722</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Trybuła, Stanisław (1961). "On the paradox of three random variables". Applicationes Mathematicae. 4 (5): 321–332. <a href="https://eudml.org/doc/264121" target="_blank">https://eudml.org/doc/264121</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021. <a href="https://www.jstor.org/stable/2690722" target="_blank">https://www.jstor.org/stable/2690722</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021. <a href="https://www.jstor.org/stable/2690722" target="_blank">https://www.jstor.org/stable/2690722</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Rump, Christopher M. (June 2001). "Strategies for Rolling the Efron Dice". Mathematics Magazine. 74 (3): 212–216. doi:10.2307/2690722. JSTOR 2690722. Retrieved 12 January 2021. <a href="https://www.jstor.org/stable/2690722" target="_blank">https://www.jstor.org/stable/2690722</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Pegg, Ed Jr. (2005-07-11). "Tournament Dice". Math Games. Mathematical Association of America. Archived from the original on 2005-08-04. Retrieved 2012-07-06. <a href="/wiki/Ed_Pegg_Jr." target="_blank">/wiki/Ed_Pegg_Jr.</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Grime, James. "Non-transitive Dice". Archived from the original on 2016-05-14. <a href="https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/" target="_blank">https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Pasciuto, Nicholas (2016). "The Mystery of the Non-Transitive Grime Dice". Undergraduate Review. 12 (1): 107–115 – via Bridgewater State University. <a href="http://vc.bridgew.edu/undergrad_rev/vol12/iss1/18" target="_blank">http://vc.bridgew.edu/undergrad_rev/vol12/iss1/18</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Grime, James. "Non-transitive Dice". Archived from the original on 2016-05-14. <a href="https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/" target="_blank">https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Reid, Kenneth; McRae, A.A.; Hedetniemi, S.M.; Hedetniemi, Stephen (2004-01-01). "Domination and irredundance in tournaments". The Australasian Journal of Combinatorics [electronic only]. 29. <a href="https://www.researchgate.net/publication/266258217" target="_blank">https://www.researchgate.net/publication/266258217</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Georgescu, Nicholas. "math_problems/intransitive.ipynb at main · NGeorgescu/math_problems". GitHub. Archived from the original on 27 March 2025. Retrieved 2025-03-27. <a href="https://web.archive.org/web/20250327230902/https://github.com/NGeorgescu/math_problems/blob/main/intransitive.ipynb" target="_blank">https://web.archive.org/web/20250327230902/https://github.com/NGeorgescu/math_problems/blob/main/intransitive.ipynb</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Grime, James. "Non-transitive Dice". Archived from the original on 2016-05-14. <a href="https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/" target="_blank">https://web.archive.org/web/20160514125245/http://grime.s3-website-eu-west-1.amazonaws.com/</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Reid, Kenneth; McRae, A.A.; Hedetniemi, S.M.; Hedetniemi, Stephen (2004-01-01). "Domination and irredundance in tournaments". The Australasian Journal of Combinatorics. 29. <a href="https://www.researchgate.net/publication/266258217" target="_blank">https://www.researchgate.net/publication/266258217</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Georgescu, Nicholas S. (2024). "Georgescu Dice - Four-Player Intransitive Solution". <a href="https://github.com/NGeorgescu/math_problems/blob/main/intransitive_5_player.ipynb" target="_blank">https://github.com/NGeorgescu/math_problems/blob/main/intransitive_5_player.ipynb</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Youhua Li (2024). "Li Dice - General n-player Extension". <a href="https://github.com/NGeorgescu/math_problems/issues/1" target="_blank">https://github.com/NGeorgescu/math_problems/issues/1</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Muñoz Perera, Adrián. "A generalization of intransitive dice" (PDF). Retrieved 15 December 2024. <a href="/w/index.php?title=Adri%C3%A1n_Mu%C3%B1oz_Perera&action=edit&redlink=1" target="_blank">/w/index.php?title=Adri%C3%A1n_Mu%C3%B1oz_Perera&action=edit&redlink=1</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> </ol>